Tu fais lire ce livre à Alex ?

Is correct.

When you want to say “make him/her do something”, you need to use the indirect pronoun “lui” instead of “le/la”, correct?

Yes, in French “lui” is equivalent to both him and her as indirect object pronouns.

Am I wrong or can’t:

Tu le fais lire à Alex.

mean both

I have Alex read it.

and

I have it read to Alex/I have someone unnamed read it to Alex ?

I have always thought a sentence like this to be ambiguous, requiring a context for its meaning to be clear.

Also, while I’m at it, may I ask whether you could use the preposition “par” here to disambiguate: “Tu le fais lire par Alex” = “I have Alex read it”?